Triangle Solver SSS – All Angles from 3 Sides

Given three sides, the solver first checks the triangle inequality — each side must be shorter than the sum of the other two — then applies the law of cosines to the largest angle. For sides 3, 4, 5: cos(C) = (9+16−25)/(2·3·4) = 0, so C = 90°, the classic right triangle. The law of sines recovers the rest: sin(A)/3 = sin(90°)/5, so sin(A) = 0.6 and A ≈ 36.87°, leaving B ≈ 53.13°. Heron’s formula finishes the picture with s = (3+4+5)/2 = 6: Area = √(6×3×2×1) = √36 = 6. The output lists every intermediate cosine and sine value, the perimeter 12, and the semi-perimeter, giving you a complete worked solution for geometry problems where only the sides are measured.

1. 1

   Enter sides a, b, c

   Any consistent unit; the triangle inequality is verified first

2. 2

   Solve the largest angle

   Law of cosines against the longest side avoids ambiguity

3. 3

   Recover the other angles

   Law of sines delivers the two remaining angles

4. 4

   Compute area & perimeter

   Heron’s formula and the side sum complete the answer

• 1

  If a² + b² = c² the triangle is right-angled at C (3²+4² = 5²)

• 2

  The largest angle always faces the longest side

• 3

  Degenerate cases (2+3 = 5) fail the triangle inequality

• Pairing the law of cosines with the wrong side — angle C must face side c

• Skipping the triangle inequality and "solving" impossible sides 1, 2, 10

• Using the sine rule to find the largest angle first, risking the ambiguous case

• Mixing Heron’s s (semi-perimeter) with the full perimeter